How do I paint (wrap) a texture around a mesh with even texture dimensions?
I would suggest with tiled patterns like this that you actually avoid manually painting until you have to deal with seams on your object. If you can use quads on your mesh, reset the UV unwrap for mapping all faces to a single tiled texture and you can then make use of Follow Active Quads to change up the mapping. Scale the UV up and down in the UV editor and you can manipulate the mapping with realtime feedback.